Modeling continuous dynamical systems from discretely sampled observations is
a fundamental problem in data science. Often, such dynamics are the result of
non-local processes that present an integral over time. As such, these systems
are modeled with Integro-Differential Equations (IDEs); generalizations of
differential equations that comprise both an integral and a differential
component. For example, brain dynamics are not accurately modeled by
differential equations since their behavior is non-Markovian, i.e. dynamics are
in part dictated by history. Here, we introduce the Neural IDE (NIDE), a novel
deep learning framework based on the theory of IDEs where integral operators
are learned using neural networks. We test NIDE on several toy and brain
activity datasets and demonstrate that NIDE outperforms other models. These
tasks include time extrapolation as well as predicting dynamics from unseen
initial conditions, which we test on whole-cortex activity recordings in freely
behaving mice. Further, we show that NIDE can decompose dynamics into their
Markovian and non-Markovian constituents via the learned integral operator,
which we test on fMRI brain activity recordings of people on ketamine. Finally,
the integrand of the integral operator provides a latent space that gives
insight into the underlying dynamics, which we demonstrate on wide-field brain
imaging recordings. Altogether, NIDE is a novel approach that enables modeling
of complex non-local dynamics with neural networks.

本文介绍了基于 Integro-Differential Equations（IDEs）理论的一种新型深度学习框架：Neural IDE（NIDE），采用神经网络学习积分算子，可对复杂的非局部动态系统进行建模，并应用于多个玩具和大脑活动数据集，验证了其优于其他模型的性能，并可将动态系统分解为马尔可夫和非马尔可夫构成部分。